Spherical distribution vectors

In this paper we consider a locally compact second countable unimodular group G and a closed unimodular subgroup H. Let ρ be a finite-dimensional unitary representation of H with closed image. For the unitary representation of G obtained by inducing ρ from H to G a decomposition in Hilbert subspaces of a certain space of distributions is given. It is shown that the representations relevant for this decomposition are determined by so-called (ρ,H) spherical distributions, which leads to a description of the decomposition on the level of these distributions